Stochastic Inference (LNA) Overview

Biochemical reactions are stochastic in nature, and the distribution of stochastic simulation trajectories is generally non-Gaussian. To meet the Gaussian noise assumption of a GP and to consider computational efficiency, we employ the linear noise approximation (LNA), a first-order expansion of the Chemical Master Equation (CME) by decomposing the stochastic process into two ordinary differential equations (ODEs); one describing the evolution of the mean and the other describing the evolution of the covariance of the trajectories $\textbf{x}(t)$ (i.e. $\mathbf{x}(t)\sim \mathcal{N}(\phi(t),\Sigma(t))$):

\[\begin{align*} \frac{d\phi(t)}{dt}&=S\cdot \mathbf{a}(\phi(t)) \label{mean} \\ \frac{d\Sigma(t)}{dt}&=S\cdot J \cdot \Sigma(t) + \Sigma(t) \cdot (J\cdot S)^T+ \Omega^{-1/2} S\cdot \mathrm{diag} \{\mathbf{a}(\phi(t))\} \cdot S^T \label{covar} \end{align*}\]

Here $S$ is the stoichiometry matrix of the system, $\textbf{a}$ is the reaction propensity vector. The $J(t)_{jk}=\partial a_j/\partial \phi_k$ is the Jacobian of the $j^{th}$ reaction with respect to the $k^{th}$ variable.

These can be solved by numerical methods to describe how $\phi(t)$ (the mean) and $\Sigma(t)$ (the covariance) evolve with time. We can then draw samples from the above (time-dependent) multivariate Gaussian distribution and obtain realizations of stochastic simulation trajectories. Those trajectories can therefore be used for stochastic simulation based ABC. Please see Examples Section for more details.

References

• Komorowski, M., Finkenstädt, B., Harper, C.V., and Rand, D.A. (2009). Bayesian inference of biochemical kinetic parameters using the linear noise approximation. BMC Bioinformatics, 10:343.

• Schnoerr, D., Sanguinetti, G., and Grima, R. (2017). Approximation and inference methods for stochastic biochemical kinetics—a tutorial review. Journal of Physics A: Mathematical and Theoretical, 50(9), 093001.